A363499 - OEIS

%I #35 Jul 20 2023 07:22:48

%S 0,1,2,3,35,67,99,131,163,406,649,892,1135,1378,1621,1864,2888,3912,

%T 4936,5960,6984,8008,9032,10056,11080,14205,17330,20455,23580,26705,

%U 29830,32955,36080,39205,42330,45455,53231,61007,68783,76559,84335,92111,99887

%N a(n) = Sum_{k=0..n} floor(sqrt(k))^5.

%C Partial sums of the fifth powers of the terms of A000196.

%H Karl-Heinz Hofmann, <a href="/A363499/b363499.txt">Table of n, a(n) for n = 0..10000</a>

%F a(n) = (n+1)*m^5 - (1/84)*m*(m+1)*(2*m+1)*(3*m-1)*(10*m^3-7*m+4), where m = floor(sqrt(n)).

%t Table[(n + 1) #^5 - (1/84) # (# + 1)*(2 # + 1)*(3 # - 1)*(10 #^3 - 7 # + 4) &[Floor@ Sqrt[n]], {n, 0, 42}] (* _Michael De Vlieger_, Jun 10 2023 *)

%o (Python)

%o from math import isqrt

%o def A363499(n): return (m:=isqrt(n))**5 *(n+1) - (m*(m+1)*(2*m+1)*(3*m-1)*(10*m**3-7*m+4))//84 # _Karl-Heinz Hofmann_, Jul 17 2023

%Y Sums of powers of A000196: A022554 (1st), A174060 (2nd), A363497 (3rd), A363498 (4th), this sequence (5th).

%K nonn,easy

%O 0,3

%A _Hans J. H. Tuenter_, Jun 05 2023